Week 03.06.2024 – 09.06.2024

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Coleman made observations about overconvergent modular forms of weight at least 2 and critical slope which imply that they are almost spanned by two subspaces corresponding to two different kinds of twist of ordinary overconvergent modular forms. He also showed that the “almost” is accounted for by a square-nilpotent action of Hecke operators. Motivated by questions about Galois representations associated to these forms, we intersect these two twists to define “bi-ordinary” forms. But we do this in a derived way: the sum operation from the two twisted ordinary subspaces to the space of critical forms defines a length 1 “bi-ordinary complex," making the bi-ordinary forms the 0th degree of bi-ordinary cohomology and realizing the square-nilpotent Hecke action as a degree-shifting action. Relying on classical Hida theory as well as the higher Hida theory of Boxer-Pilloni, we interpolate this complex over weights. We can deduce “R=T” theorems in the critical and bi-ordinary cases from R=T theorems in the ordinary case. And specializing to weight 1 under a supplemental assumption, we show that the bi-ordinary complex with its square-nilpotent Hecke action specializes to weight 1 coherent cohomology of the modular curve with a degree-shifting action of a Stark unit group. The action is a candidate for a p-adic realization of conjectures about motivic actions of Venkatesh, Harris, and Prasanna. This is joint work with Francesc Castella.

Cohomology is the most fundamental global invariant attached to a sheaf. For an l-adic local system L on the complement of a divisor D in a smooth projective variety over an algebraically closed field of characteristic p not equal to l, we will advertise the existence of estimates for the rank of each cohomology spaces of L depending only on local data : the rank of L and the ramification conductors of L at the generic points of D. This is joint work with Haoyu Hu.

Using the notion of integral distance to analytic functions, we give a characterization of Schatten class Hankel operators acting on doubling Fock spaces on the complex plane and use it to show that for $f\in L^{\infty}$ if $H_{f}$ is Hilbert-Schmidt, then so is $H_{\bar{f}}$. This property is known as the Berger-Coburn phenomenon. When $0 < p \le 1$, we show that the Berger-Coburn phenomenon fails for a large class of doubling Fock spaces. Along the way, we illustrate our results for the canonical weights $|z|^{m}$ when $m > 0$.